# Misc 26 - Chapter 9 Class 11 Sequences and Series

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Misc 26 Show that (1 22 + 2 32 + + n (n + 1)2)/(12 2 + 22 3 + + n2 (n + 1)) = (3n + 5)/(3n + 1) Taking L.H.S (1 22 + 2 32 + + n (n + 1)2)/(12 2 + 22 3 + + n2 (n + 1)) We solve denominator & numerator separately Solving numerator Let numerator be S1 = 1 22 + 2 32 + + n (n + 1)2 nth term is n (n + 1)2 Let an = n(n + 1)2 = n(n2 + 1 + 2n) = n3 + n + 2n2 Now finding S1 = (( ( + 1))/2)^2 + 2(( ( +1)(2 +1))/6) + n(n+1)/2 = ( ( + 1))/2 (n(n+1)/2 " + " (2(2 +1))/3 " + 1" ) = ( ( + 1))/2 (( 3 ( +1) + 2 2(2 +1)+ 6)/6) = (n(n + 1))/(2 6)[3n(n + 1) + 4(2n + 1) + 6] = (n(n + 1))/12[3n2 + 3n + 8n + 4 + 6] = ( ( + 1))/12[3n2 + 11n + 10] = ( ( + 1))/12[3n2 + 5n + 6n + 10] = ( ( + 1))/12[n(3n + 5) + 2(3n + 5)] = ( ( + 1))/12[(n + 2)(3n + 5)] Thus, S1 = ( ( + 1))/12[(n + 2)(3n + 5)] Now solving denominator Let denominator be S2 = 12 2 + 22 3 + + n2 (n + 1) nth term is n2(n + 1) Let bn = n2(n + 1) bn = n3 + n2 Now, calculating S2 = (( ( + 1))/2)^2 + (( ( +1)(2 +1))/6) = ( ( + 1))/2 (n(n+1)/2 " + " ((2 +1))/3) = ( ( + 1))/2 (n(n+1)/2 " + " ((2 +1))/3) = ( ( + 1))/2 (( 3 ( +1) + 2 (2 +1))/6) = (n(n + 1))/(2 6) (3n(n + 1) + 2(2n + 1)) = (n(n + 1))/12 (3n2 + 3n + 2(2n + 1)) = (n(n + 1))/12 (3n2 + 3n + 4n + 2) = (n(n+1))/12 (3n2 + 7n +2) = (n(n+1))/12 (3n2 + 6n + n +2) = (n(n+1))/12 (3n(n + 2) + 1(n +2)) = (n(n+1)(n+2)(3n+1))/12 Thus, S2 = (n(n+1)(n+2)(3n+1))/12 Now, Taking L.H.S (1 22 + 2 32 + + n (n + 1)2)/(12 2 + 22 3 + + n2 (n + 1)) = 1/ 2 = ((n(n+1)(n+2)(3n+5))/12)/((n(n+1)(n+2)(3n+1))/12) = (n(n+1)(n+2)(3n+5))/12 12/(n(n+1)(n+2)(3n+1)) = (n(n+1)(n+2)(3n+5))/(n(n+1)(n+2)(3n+1)) = ((3n+5))/((3n+1)) = R.H.S Hence L.H.S = R.H.S Hence proved.

Miscellaneous

Misc 1

Misc 2

Misc 3 Important

Misc 4

Misc 5

Misc 6

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11

Misc 12

Misc 13

Misc 14

Misc 15

Misc 16 Important

Misc 17

Misc 18

Misc 19 Important

Misc 20

Misc 21

Misc 22

Misc 23

Misc 24

Misc 25 Important

Misc 26 You are here

Misc 27

Misc 28 Important

Misc 29

Misc 30

Misc 31

Misc 32 Important

Chapter 9 Class 11 Sequences and Series

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.